2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal,
Electron. J. Diff. Eqns., Conference 22 (2015), pp. 63-77.

Remarks and examples on transient processes and attractors in biological evolution

Philippe Lherminier, Evariste Sanchez-Palencia

Abstract:
We present a model for the competition of two biological entities into the same species (polyphasie, clonal/sex, cancerous cells), the first one with a birth ratio higher than the second when the resources are abundant, whereas the situation is reversed for scarce resources. The first one rapidly exhausts the resources, improving growth of the second, leading to a auto-sustained cyclic process (ESS = Evolutionary Stable Strategy). We use known models of population dynamics for three agents: two phases asexual and sexual (for instance) of the same species and one of resources. The main feature of the model (for certain values of the parameters) is the very long and entangled transient process, which involves a long period where one of the forms is practically absent, before emerging again to join a stable cycle which implies preservation of both forms. This model should throw some light on the biological problem of the maintenance of sexuality in competition with asexual clones, as well as on the alternated fast growth versus latency in cancer tumors.

Published November 20, 2015.
Math Subject Classifications: 34C05, 34A34, 34C14
Key Words: Darwinian dynamics; emergence of structured patterns; maintenance of sex.

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Philippe Lherminier
Château du Fontenil
61300 Saint Sulpice, France
email: phlherminier@wanadoo.fr
Evariste Sanchez-Palencia
Sorbonne Universités, UPMC Univ Paris 06
and CNRS, UMR 7190, Institut Jean Le Rond d'Alembert
F-75005, Paris, France
email: sanchez@dalembert.upmc.fr

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